the system, taken with respect to the same axis, by r the principle of moments (Art. 28),

(P+P' + P' + &c.)r = Pp + P'p' + &c.;

From

(12.)

Hence, the lever arm of a system of parallel forces, taken with respect to an axis at right-angles to their direction, is equal to the algebraic sum of the moments of the forces divided by the algebraic sum of the forces.

Centre of Parallel Forces.

43. Let there be any number of forces, P, P', P'', &c., applied at points invariably connected together, and whose co-ordinates are x, y, z; x', y', z' ; x'', y'', z''; &c. Let R denote their resultant, and represent the co-ordinates of its point of application, by 2, y, and z,; denote the angles made by the common direction of the forces with the axes of X, Y, and Z, by a, B, and y.

Suppose each force resolved into three components, respectively parallel to the co-ordinate axes, the points of application being unchanged:

The components parallel to the axis of X are,

Pcosa, P'cosa, P''cosx, &c., Rcosa;

those parallel to the axis of Yare,

Pcosß, P'cos3, P'cosß, &c., Reosß;

and those parallel to the axis of Z are,

Peosy, P'cosy, P'cosy, &c., Reosy.

If we take the moments of the components parallel to the axis of Z, with respect to the axis of Y, as an axis of moments, we shall have, for the lever arms of the components, x, x', x', &c.; and from the principle of moments (Art. 36),

Reosyx, Pcosy x + P'cosy x' + &c.

=

Striking out the common factor cos y, and substituting for R its value, we have,

In like manner, if we take the moments of the same components, with respect to the axis of X, we shall have,

And, if we take the moments of the components parallel to the axis of Y, with respect to the axis of X, we shall have,

Hence we have for the co-ordinates of the point of application of the resultant,

These co-ordinates are entirely independent of the direc tion of the parallel forces, and will remain the same so long as their intensities and points of application remain unchanged.

The point whose co-ordinates we have just found, is called the centre of parallel forces.

Resultant of a Group of Forces in a Plane, and applied at points invariably connected.

44. Let P, P', P", &c., be any number of forces lying in the same plane, and applied at points invariably connected together; that is, at points of the same solid body.

Through any point O in the plane of the forces, draw any

with the axis OY, by B, B', B", &c.; denote, also, the coordinates of the points of application of the forces, by x, y; x', y'; x", y"; &c.

Let each force be resolved into components parallel to the co-ordinate axes; we shall have for the group parallel to the axis of X,

Pcosz, P'cosa', P'cosx", &c.;

and, for the group parallel to the axis of Y,

Pcos, P'cos', P'cos", &c.;

The resultant of the first group is equal to the algebraic sum of the components (Art. 38); denoting this by X, we shall have,

In like manner, denoting the resultant of the second group by Y, we shall have,

The forces X and Y intersect in a point, which is the point of application of the system of forces. Denoting the resultant by R, we shall have (Art. 33),

To find the point of application of R, let O be taken as a centre of moments, and denote the lever arms of X and Y

by y, and 2, respectively. From the principle of Article

If we denote the angles which the resultant makes with the axes of X and Y by a and b respectively, we shall have, as in Article 33,

Equations (16) and (17) make known the point of application, and Equations (18) make known its direction; hence, the resultant is completely determined.

To find the moment of R, with respect to O as a centre of moments, let us denote its lever arm by r, and the lever arms of P, P', P'', &c., with respect to O, by p, p', p'', &c. The moment of the force Pcosa, is Pcosa y, and that of the force Pcos3, is Peosßx. The negative sign is given to the last result, because the forces Pcosa and Pcos tend to turn the system in contrary directions.

From the principle of moments (Art. 36), the moment of P is equal to the algebraic sum of the moments of its components. Hence,

In like manner, the moments of the other component forces may be found. Because the moment of the resultant is equal to the algebraic sum of the moments of all its components (Art. 36), we have,

Rr=

(Pp) = (Pcosa y - Pcos x) (19.)

Resultant of a Group of Forces situated in Space, and applied at points invariably connected.

Ꮓ

Prosp

45. Let P, P', P", &c., be any number of forces situated in any manner in space, and applied at points of the same solid body. Assume any point in space, and through it draw any three lines perpendicular to each other. Assume these lines as axes. Denote the angles which the forces P, P', P'', &c., make with the axis of X, by a, a', a", &c.; the angles which they make with the axis

0

Pcosy

X

Fig. 29.

Pcosa

of Y, by 8, 8', B", &c.; the angles which they make with the axis of Z, by y, y', y', &c., and denote the co-ordinates of their points of application by x, y, z; x', y', z'; x", y", 2"; &c.

Let each force be resolved into components respectively parallel to the co-ordinate axes.

We shall have for the group parallel to the axis of X,

Pcosa, P'cosa', P'cosz", &c.;

for the group parallel to the axis of Y,

Pcosß, P'cos', P''cosß", &c.;

and for the group parallel to the axis of Z,

Pcosy, P'cosy', P'cosy", &c.

Denoting the resultants of these several groups by X, Y, and Z, we shall have,

X = (Peos,) Y= (Pcos,) and Z = (Pcosy) (20.)

These three forces must intersect at a point, which point is the point of application of the resultant of the entire sys